Related papers: Splines in geometry and topology
Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which are stitched together through transition maps. They are called $G^{r}$-splines if, after composition with the transition maps, they…
Issues relevant to the flow chirality and structure are focused, while the new theoretical results, including even a distinctive theory, are introduced. However, it is hope that the presentation, with a low starting point but a steep rise,…
This article introduces a new term "splint" and classifies the splints of the classical root systems. The motivation comes from representation theory of semisimple Lie algebras. In a few instances, splints play a role in determining…
Symmetries and isomorphisms play similar conceptual roles when we consider how models represent physical situations, but they are formally distinct, as two models related by symmetries are not typically isomorphic. I offer a rigorous…
We consider various equivalence relations on the set of homotopy classes of curves on a hyperbolic surface based on topological, algebraic, and geometric structures. The purpose of this work is to determine the relationship between these…
We discuss possible relationships between geometric and topological interactions on one side and physical interactions on the other side.
We determine the symmetrized topological complexity of the circle, using primarily just general topology.
This article introduces spotlight tiling, a type of covering which is similar to tiling. The distinguishing aspects of spotlight tiling are that the "tiles" have elastic size, and that the order of placement is significant. Spotlight…
The purpose of this thesis is to use the language of orbifold groupoids to describe the geometry and topology of orbifolds, highlighting advantages and disadvantages of this language as they arise.
In this study, a new form of quadratic spline is obtained, where the coefficients are determined explicitly by variational methods. Convergence is studied and parity conservation is demonstrated. Finally, the method is applied to solve…
This is a survey on stated skein algebras and their representations.
This paper proposes a simple technique of curve and surface construction with B-splines. Given a control polygon or a control mesh together with node ordinates corresponding to all control points, a rational curve or surface is obtained by…
We propose a new type of state sum model for two-dimensional surfaces that takes into account topology and spin. The definition used - new to the literature - provides a rich class of extended models called spin models. Both examples and…
The main aim of this article is to introduce sk-splines on R^n and establish representations of cardinal sk-splines with knots and points of interpolation on the sets AZ^n, where A is an arbitrary nonsingular matrix. Such sets of points are…
Covering-based rough set theory is an extension to classical rough set. The main purpose of this paper is to study covering rough sets from a topological point of view. The relationship among upper approximations based on topological spaces…
This expository talk is an expanded version of a lecture at G.-M. Greuel's 60th Birthday Conference in Kaiserslautern in October, 2004. We survey recent work of Neumann-Wahl and others on the relation between topology and geometry of normal…
In this paper, we overview one promising avenue of progress at the mathematical foundation of deep learning: the connection between deep networks and function approximation by affine splines (continuous piecewise linear functions in…
Spherical spin glasses are canonical models for smooth random functions in high dimensions. In this review, we survey several interrelated lines of research on their geometric structure. We begin with results concerning critical points and…
We study the notion of geometric structures for toposes: This generalizes the notion of (X,G) manifolds. We give some applications to algebraic geometry
Engineering design via CAD software relies on Non-Uniform Rational B-Splines (NURBS) as a means for representing and communicating geometry. Therefore, in general, a NURBS description of a given design can be considered the exact…